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The points where a function's graph crosses the x-axis (where $y = 0$) can be referred to as zeros, x-intercepts, roots, or solutions.

Conditions: The function is being viewed as a graph in the coordinate plane.; The context is solving $f(x) = 0$.

The points where a function's graph crosses the x-axis (where $y = 0$) can be referred to as zeros, x-intercepts, roots, or solutions.

Conditions: The function is being viewed as a graph in the coordinate plane.; The context is solving $f(x) = 0$.

To find an interval that must contain a zero, inspect the signs of the y-values in the table for consecutive x-values. If the sign changes from positive to negative or vice versa between two x-values, the Intermediate Value Theorem guarantees a zero in that interval for a continuous function.

Conditions: The function is treated as continuous.; The table contains discrete sample points of the function.

To find an interval that must contain a zero, inspect the signs of the y-values in the table for consecutive x-values. If the sign changes from positive to negative or vice versa between two x-values, the Intermediate Value Theorem guarantees a zero in that interval for a continuous function.

Conditions: The function is treated as continuous.; The table contains discrete sample points of the function.

The Intermediate Value Theorem only guarantees a zero if the function changes sign (from positive to negative or vice versa) over an interval. If both y-values are positive or both are negative, a zero is not guaranteed because the function might stay above or below the axis without crossing it.

Conditions: The function is continuous.; The y-values at the endpoints of the interval have the same sign.

The Intermediate Value Theorem only guarantees a zero if the function changes sign (from positive to negative or vice versa) over an interval. If both y-values are positive or both are negative, a zero is not guaranteed because the function might stay above or below the axis without crossing it.

Conditions: The function is continuous.; The y-values at the endpoints of the interval have the same sign.